Hello amazing math people. We're going to do multiplying polynomials. We're going to start with two monomials which just have a coefficient and a variable. Let's say 6 X squared yz to the 7th times X ^3 Y 4th Z. When we multiply polynomials, we're going to multiply the coefficients. This is a six, and if there's not a number, it's understood to be a 1, so 6 * 1 would give us 6. When we multiply the variable portion, we're going to add the exponents. So if I have X ^2 * X ^3, 2 + 3 is going to give us X to the 5th When we do the same thing to the Y, If there's no number, it's an understood 1. So Y to the 1 + 4, or again we're going to get 5. Finally, looking at the Z portion, Z to the 7th times Z, we're going to add 7 plus that understood one and get Z to the 8th if it happens to be there. There weren't any variables of a certain type, such as three X ^2 y * 5 AX. We're just going to multiply the coefficients 3 * 515. There's a single A and there's nothing to add the exponents with because there was only 1A, the X's. We're going to add those exponents. Here's an understood 2 and an understood 1. So 2 + 1's going to give us 3, and the Y didn't have anything to combine with it either. Now, once we do multiplying monomials, we can build to multiplying polynomials. If I have something in front, a monomial times something that's not a monomial, let's say times a trinomial, we're going to take this monomial in front and multiply it by each and every piece of the polynomial. So we're going to do the distributive property that 6X is going to get multiplied by the two X squared by the 3X and by the -1. So now we're going to use the skills we just talked about up here. 6 * 2 is 12X times X ^2 is X ^3. 6 * 3 is 18X times X. Remember, we add the exponents, so 1 + 1 is 2 and then six X * -1 -, 6 X. Now we can expand that even further. Let's say that we have three X + 2 * 4 X -5. Now what we're going to do is we're going to think about taking this first term here 3X and multiplying it by that whole four X -, 5. So that 3X is going to get multiplied by the whole four X -, 5. Then we're going to take the two and we're going to multiply it by that four X -, 5. Now the skill we just talked about, we're going to distribute. So we're going to have the 3X times the 4X to get 12 X squared and the 3X times the -5 negative 15X, the two times the 4X8X, and the two times the -5 negative 10. If we combine our like terms, we'd get twelve X ^2 -, 7 X -10. This sometimes can be thought of as a shortcut method called FOIL. FOIL only works if it's a binomial 2 terms times 2 terms. And what the FOIL method does is it says take these first terms and multiply them together the three X times the 4X. Then take the outer terms and multiply them together 3X times -5. If you look up here, that's exactly what we did, 3X times 4X and three X * -5. Then we're going to do the inner terms, the two times, the 4X. And then we're going to do the last terms, the two times the -5 So we're going to do the two times the 4X and the two times the -5 right here, 2 times, the 4X2 times the -5. Now, if you squint and you have a pretty good imagination, you could actually think about that looking a little bit like Charlie Brown. And if we did the foil, the first terms give me the 12 X squared, the outer terms -15 X the inner terms 8X and the last terms -10 we combine our like terms twelve X ^2 -, 7 X -10. We get the same answer either method we want to use. We have a couple special types of equations. 1 is called squaring of a binomial and what we do is we actually start with a binomial A+B and we square it. So a + b ^2 just means A+B times itself. And if we do our FOIL method, our first terms A times a a ^2 outer A * B or AB inner AB again last b ^2. We combine our like terms a ^2 + 2 AB plus b ^2. This formula holds also if it was subtraction, so we'd have A -, b ^2. The only difference is that the two AB would become A -, 2 AB. The last special equation that I want to talk about at this moment is called the product of a sum and difference. If I have a + b * a -, b, when I do my foil here, my first terms a ^2, my outer terms negative AB, my inner terms positive AB, my last terms negative b ^2. Once again, there's Charlie Brown smiling at you. The negative AB and positive AB cancel and we get a ^2 -, b ^2. This is the product of a sum and a difference. You guys are doing great, have a nice day.